We know that: $\Delta u=-f, \in\Omega$ $u=0$ ,on $\partial\Omega$
Is equivalent to this variational problem:
find the minimum of $J(u)$ in $C^2(\Omega)\cap C^1(\bar\Omega)$, where $J(u)=\int_\Omega[\frac{1}{2}(u_x^2+u_y^2)-uf]dxdy$.
My question is:
why we need u belongs to $C^2(\Omega)\cap C^1(\bar\Omega)$? specifically, why we need $u,Du$ continuos up to the boundary of $\Omega$? Why can't we just use $u\in C^2(\Omega)\cap C^1(\Omega)$?
Typically the requirement is $u\in C^2(\Omega)\cap C^0(\bar{\Omega})$ if you want a classical solution; you need this because you have boundary conditions and you want u to be continuous on the boundary (otherwise you could just find a solution without BC in the interior and define it as $0$ on the boundary). The (stronger) requirement $u\in C^2(\Omega)\cap C^1(\bar{\Omega})$ is a bit weird actually, I've never heard about it, but it guarantees continuity at $\partial\Omega$ . However it may be useless.